·· P :p :: m cos. 3 x-n cos.λ: n sin.2 a — m sin.2 λ'

ora: b:: (m cos.'x' -n cos.λ): (n sin. x — m sin.' x')

Q. E. I.

Again, let L' and ' denote the given lengths of two pendulums vibrating seconds in the latitudes λ, x'; then since the length of the pendulum c force, and by Simpson's Fluxions, Vol. II. Art. 388, it is shewn that the attraction upon a spheroid ∞ that part of normal to the generating ellipse passing through the body attracted, which is intercepted between the ellipse and axis-minor, there

fore

But

L : force at equat.: ƒ. at pole

::b: a.

L-L-L:: sin. sin.x'

.. L' sin.'x' L sin.'x'=' sin. x L sin."λ

:. L =

Also

=

sin.

(l' sin. L'sin.x') cos.x

sin. a (sin. a - sin.' x')

L'sin.-L'sin.'x'—l'sin.x+lsin.*+Lsin.'x'—L'sin.*asin."'

sin. (sin.asin. x')

ba:: L:1:: l' sina - L' sin.x: L'cos.l'cos. a

the ratio required.

625.

Generally we have (see Jesuit's edition of Newton,

note to cor. 2. prop. LIII.)

where F denotes the required force, f the force corresponding to a given arc to be described, viz., b, p the on the tangent, and g the radius vector. Now in the logarithmic spiral,

a

P=P sin. a,

being the constant angle between the radius vector and curve,

but when Ff, let g = R; then cb cos.a – R

.. F =

.. Fas

the law required.

626.

We know that the time of an oscillation is generally

But since the pendulum when on the top of the mountain loses n seconds in a day, the time of one vibration is

627.

Let T be the time of vibration at the pole; then the

time of vibration at the other place is

F being the accelerating force, and L the given length of the pendulum. But by 621

R being the radius of the sphere, a the given latitude, and t the

1′′ and 1”. Hence

630.

This curve is called the Tractrix, and its equa

tion is

ydx=dy √ (a2 — y3)

(see Whewell's Dynamics, p. 127,) y, x, being the ordinate and abscissa originating in B, and a = AB.

631.

Let a be the length of the chain, w its weight, b the part unwound at the commencement of the motion, r the radius of the quadrant, and u the part unwound at any time during the descent; then since the accelerating force upon every particle du = ds of the chain in contact with the curve, by the theory of the inclined plane, is (y and s being the ordinate and arc of the quadrant,)